Find the value of `(1)/( 3) + ( 1)/( 15) + ( 1)/( 35) + ( 1)/( 63) + ( 1)/ ( 99)`

Options

A. `( 10)/( 11)`
B. `( 5)/( 11)`
C. `( 9 )/( 11)`
D. `( 7 )/( 11)`

Answer

To find the value of \( \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} \), we will follow these steps:

### Step 1: Identify the fractions
We have the following fractions to add:
\[ \frac{1}{3}, \frac{1}{15}, \frac{1}{35}, \frac{1}{63}, \frac{1}{99} \]

### Step 2: Find a common denominator
To add these fractions, we need a common denominator. The denominators are 3, 15, 35, 63, and 99. The least common multiple (LCM) of these numbers will be our common denominator.

Calculating the LCM:
- The prime factorization of each number is:
- \(3 = 3^1\)
- \(15 = 3^1 \times 5^1\)
- \(35 = 5^1 \times 7^1\)
- \(63 = 3^2 \times 7^1\)
- \(99 = 3^2 \times 11^1\)

The LCM will take the highest power of each prime:
- \(3^2\) from 63 and 99
- \(5^1\) from 15 and 35
- \(7^1\) from 35 and 63
- \(11^1\) from 99

Thus, the LCM is:
\[ LCM = 3^2 \times 5^1 \times 7^1 \times 11^1 = 9 \times 5 \times 7 \times 11 = 3465 \]

### Step 3: Rewrite each fraction with the common denominator
Now we rewrite each fraction with the common denominator of 3465:
\[ \frac{1}{3} = \frac{1155}{3465}, \quad \frac{1}{15} = \frac{231}{3465}, \quad \frac{1}{35} = \frac{99}{3465}, \quad \frac{1}{63} = \frac{55}{3465}, \quad \frac{1}{99} = \frac{35}{3465} \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ \frac{1155 + 231 + 99 + 55 + 35}{3465} \]

Calculating the numerator:
\[ 1155 + 231 = 1386 \]
\[ 1386 + 99 = 1485 \]
\[ 1485 + 55 = 1540 \]
\[ 1540 + 35 = 1575 \]

So, we have:
\[ \frac{1575}{3465} \]

### Step 5: Simplify the fraction
Now we simplify \( \frac{1575}{3465} \):
Finding the GCD of 1575 and 3465:
- The prime factorization of 1575 is \( 3^2 \times 5^2 \times 7^1 \).
- The prime factorization of 3465 is \( 3^2 \times 5^1 \times 7^1 \times 11^1 \).

The GCD is:
\[ 3^2 \times 5^1 \times 7^1 = 315 \]

Now, divide both the numerator and denominator by 315:
\[ \frac{1575 \div 315}{3465 \div 315} = \frac{5}{11} \]

### Final Answer
Thus, the value of \( \frac{1}{3} + \frac{1}{15} + \frac{1}{35} + \frac{1}{63} + \frac{1}{99} \) is:
\[ \frac{5}{11} \]